0$ and $n>n_{\varepsilon},$ we have:
$${\frac {\pi(n)-k(n)} {\pi(n)}}\geq {\frac {\pi(n)- \pi(n/2)} {\pi(n)}}\geq\frac{1}{2}-\varepsilon.$$
\end{proof}
Unfortunately, the analysis of the sieves obtained in section 7 seems much more difficult than the analysis of the sieve of Eratosthenes for primes. Nevertheless, some very simple probabilistic arguments lead to a very plausible conjecture about the density of over-Ramanujan and over-Labos primes. First of all, let us show that events $R':$ ``a prime is over-Ramanujan" and $L':$ ``a prime is over-Labos" are independent.
\begin{proof}Indeed, denoting events $r:$ ``a prime is right", $l:$ ``a prime is left" and $Is:$ ``a prime is isolated", we have
\begin{equation}\label{9.4}
\mathrm{P}[R' | L']=1-\mathrm{P}[l]-\mathrm{P}[Is]; \;\mathrm{P}[R' |\overline{L'} ]=1-\mathrm{P}[r]-\mathrm{P}[Is].
\end{equation}
Hence, in view of $\mathrm{P}[l]=\mathrm{P}[r]$ (cf. Proposition 5), we have
\begin{equation}\label{9.5}
\mathrm{P}[R' | L']=\mathrm{P}[R' |\overline{L'} ].
\end{equation}
\end{proof}
\begin{conjecture}\label{con1}
\begin{equation}\label{9.1}
\pi_{R'}(x)\sim (1-e^{-1})\pi(x)=0.63212\cdots\pi(x).
\end{equation}
\end{conjecture}
The following proof is heuristic.
\begin {proof} Consider asymptotically $\frac{\pi(n)}{2}$ intervals of the form $(2p_m,\; 2p_{m+1})$ covering all $\pi(n)$ primes. It is well known (\cite{5}) that, for large $n,$ an interval between two random consecutive primes on the average has length $\ln p_n.$ Thus a random interval of the considered form has length $2\ln p_n$ and, according to the Cram\'{e}r model, the number of primes in such a random interval has the binomial $(2\ln p_n,\; \frac {1} {\ln p_n})$ distribution which, for large $\ln p_n,$ has a good approximation by a Poisson distribution with parameter $\lambda=2.$ Thus we accept that a random interval contains $k$ primes with probability $\mathrm{P}[X=k]=\frac {2^k}{k!}e^{-2},\; k=0,1,2\ldots\; .$ Since $\mathrm{P}[X=0]=e^{-2},$ then the number of intervals containing at least one prime is about $\frac{\pi(n)}{2}(1-e^{-2}).$
This number corresponds to our condition, since we consider \slshape only \upshape such intervals. Furthermore, since $\mathrm{P}[X=1]=2e^{-2},$ then the probability that such an interval contains an only prime is $2e^{-2}/(1-e^{-2})$ and, consequently, we have about $\frac{\pi(n)}{2}(1-e^{-2})\cdot\frac {2e^{-2}}{(1-e^{-2})}=\pi(n)e^{-2}$ intervals
containing, by our terminology, isolated primes and this number coincides with the number of isolated primes. This means that the probability that a prime is isolated is $e^{-2}.$ On the other hand, this probability equals $\mathrm{P}[\overline{R'} \cdot\overline{L'}]=(\mathrm{P}[\overline{R'}])^2.$ Therefore, $\mathrm{P}[\overline{R'}]=e^{-1}$ and $\mathrm{P}[R']=1-e^{-1}$ which justifies the conjecture.
\end{proof}
Greg Martin \cite{4} did the corresponding calculations for the first million primes $p$, and found that approximately $61.2\% $ of them have a prime in the interval $(p, 2p_{n+1}).$ Since in this case $\ln p_n$ is small (less than 17), an error of about $2\% $ is quite acceptable.
\begin{remark}\label{r1}
It could be done also the following simple explanation of appearance of the constant $1-1/e$ in Conjecture $\ref{con1}.$ Consider a random prime $p.$ Let it be in interval $(2p_n, 2p_{n+1}).$ We accept that $p$ could be to the left or to the right from the midpoint $p_n+p_{n+1}$ with the same frequency. Thus the mathematical expectation of the distance between $p$ and $2p_{n+1}$ is the difference $p_{n+1}-p_n$ which in average is $\ln n.$ Accepting for large $n$ that the frequency of appearance a prime approximately is $\frac{1}{\ln n},$ we see that it is natural to accept that the frequency of the appearance of a prime to the right from $p$ in the considered interval is close to $1-(1-\frac{1}{\ln n})^{\ln n}$ which for large $n$ is close to $1-e^{-1}.$ Thus the frequency that a random $p$ is over-Ramanujan is $1-e^{-1}.$
\end{remark}
Note that, if Conjecture \ref{con1} is true, then, using (\ref{1.2}), for the counting function $\pi_{R^*}(x)$ of pseudo-Ramanujan primes we have
\begin{equation}\label{9.2}
\pi_{R^*}(x)\sim (\frac {1}{2}-\frac {1}{e})\pi(x)=0.13212...\pi(x),
\end{equation}
so that the proportion of Ramanujan primes among all over-Ramanujan primes is approximately 0.79099. Using Theorem \ref{t3}, we note that, if Conjecture \ref{con1} is true, then, for the counting function $\pi_{L'}(x)$ of over-Labos primes we have
\begin{equation}\label{9.3}
\pi_{L'}(x)\sim \pi_{R'}(x)\sim (1-e^{-1})\pi(x).
\end{equation}
Therefore, if Conjecture \ref{con1} is true, then, for the counting functions $\pi_l(x),\;\pi_r(x),\pi_c(x)$ and $\pi_{is}(x) $ of the left, right, central and isolated primes, respectively, of our classes of primes, we have
\begin{equation}\label{9.6}
\pi_l(x)\sim\pi_r(x)\sim (1-e^{-1})e^{-1}\pi(x)=0.2325\cdots\pi(x),
\end{equation}
\begin{equation}\label{9.7}
\pi_c(x)\sim(1-e^{-1})^2\pi(x)=0.3995\cdots\pi(x),
\end{equation}
\begin{equation}\label{9.8}
\pi_{is}(x)\sim e^{-2}=0.1353\cdots\pi(x),
\end{equation}
so that $\pi_r(x)+\pi_l(x)+\pi_c(x)+\pi_{is}(x)=\pi(x).$
\section{A generalization}
Let us consider a natural generalization of Ramanujan primes.
\begin{definition}\label{d7}
For a real $v>1,$ a \emph{$v$-Ramanujan prime} is the largest prime $(R_v(n))$ for which $\pi(R_v(n))-\pi(R_v(n/v))=n.$
\end{definition}
As in case $v=2,$ equivalently $R_v(n)$ is the \emph{smallest integer with the property: if $x \geq R_v(n),$ then} $\pi(x) - \pi(x/v) \geq n.$ {\tt}
Note that, evidently,
\begin{equation}\label{10.1}
R_v(n)\sim p_{((v/(v-1))n)}
\end{equation}
as $\rightarrow\infty.$ Let $\pi_R^{(v)}(x)$ be the counting function of $v$-Ramanujan primes. Then (cf. (\ref{1.2}))
\begin{equation}\label{10.2}
\pi_R^{(v)}(x)\sim(1-1/v)\pi(x).
\end{equation}
Put
\begin{equation}\label{10.3}
\kappa(v)=\begin{cases}0,&
\text{if $v$ is not the ratio between two primes}; \\
r,& \text{if $v=\frac{r}{q}$ where $r$ and $q$ are primes}.
\end{cases}
\end{equation}
The following theorem is proved in the same way as Theorem 1.
\begin{theorem}\label{t5}
Let $v>1$ be a given real number. If $p>\max(2v,\; \kappa(v))$ is $v$-Ramanujan prime such that $p_m< p/v< p_{m+1},$ then the interval $(p,\; \lceil vp_{m+1}\rceil+\varepsilon)$ contains a prime.
\end{theorem}
\begin{remark}\label{r2}
The condition $p>\max(2v,\; \kappa(v))$ allows us to avoid the cases $p=2v$ and $p=vq$ with a prime $q,$ when the condition $p_n< p/v< p_{n+1}$ is impossible.
\end{remark}
Let us find an upper bound on the $n$-th $v$-Ramanujan prime.
\begin{theorem}\label{t6} If $n\geq\frac{1}{k}\max (6k,\; e^v,\; v^{(0.79677\frac{k-1}{k}v-1)^{-1}}),$ then, for $v\geq1.25507\frac{k}{k-1},$ we have
\begin{equation}\label{10.4}
R_v(n)\leq p_{kn}.
\end{equation}
\end{theorem}
\begin{proof} It is sufficient to show that $\pi(\frac{p_{kn}}{v})\leq(k-1)n.$ Indeed, then we have
$\pi(p_{kn})-\pi(\frac{p_{kn}}{v})\geq kn-(k-1)n=n.$ We use the following known results (\cite{1}, \cite {9}-\cite{10}):
\begin{equation}\label{10.5}
p_nn\ln n;
\end{equation}
\begin{equation}\label{10.7}
\pi(x)<1.25506 \frac {x} {\ln x},\; x>1.
\end{equation}
Note that $\frac {p_{kn}}{v}>\frac {kn}{v}>\frac{kn}{e^v}.$ Hence, by the condition, $\frac{p_{kn}}{v}>1.$
By (\ref{10.5})-(\ref{10.7}),
$$\pi(\frac{p_{kn}}{v})<1.25506\frac {p_{kn}}{v \ln (\frac {p_{kn}}{v})}<
1.25506\frac{kn}{v}\cdot\frac {\ln(kn)+\ln(\ln(kn))}{\ln (\frac {kn\ln(kn)}{v})}$$
$$=1.25506\frac{kn}{v}(1+\frac {\ln v}{\ln (\frac {kn\ln(kn)}{v})}).$$
Taking into account that, by the condition, $\ln (kn)>v,$ we have
$$\pi(\frac{p_{kn}}{v})<1.25506\frac{kn}{v}(1+\frac {\ln v}{\ln (kn)}).$$
Finally, note that, by the condition, $\frac {\ln v}{\ln (kn)}\leq 0.7968\frac{k-1}{k}v-1.$ Therefore,
$$\pi(\frac{p_{kn}}{v})<1.25506\cdot0.79677(k-1)n\max(2v,\; \kappa(v))$ is a \emph{$v$-over-Ramanujan prime} if, as soon as $p_m< p/v< p_{m+1},$ the interval $(p,\; vp_{m+1})$ contains a prime.
\end{definition}
\begin{definition}\label{d9}
A $v$-over-Ramanujan not $v$-Ramanujan prime is a \upshape $v$-pseudo-Ramanujan prime.
\end{definition}
\emph{Now $v$-Labos primes, $v$-over-Labos primes and $v$-pseudo-Labos primes are introduced quite symmetrically} (see Section 3). In particular, the following statements are valid.
\begin{theorem}\label{t7}
Let $v>1$ be a given real number. If $p>\max(2v,\; \kappa(v))$ is $v$-Labos prime, such that $p_m< p/v< p_{m+1},$ then the interval $(\lfloor vp_m\rfloor-\varepsilon,\; p)$ contains a prime.
\end{theorem}
\begin{theorem}\label{t8}
For the sequences $\{R'_v(n)\}$ and $\{L'_v(n)\}$ of $v$-over-Ramanujan and $v$-over-Labos primes, we have
\begin{equation}\label{10.11}
R'_v(1)\leq L'_v(1)\leq R'_v(2)\leq L'_v(2)\leq\cdots
\end{equation}
\end{theorem}
A generalization of the first sieve for $v$-over-Ramanujan primes, $v\geq2,$ is based on the Bertrand-like sequence $\{b_v(n)\},$ defined by $ b_v(1)=2,$ and, for $n\geq2,$ as the largest prime less than $\lceil vb_v(n-1)\rceil+\varepsilon.$ A generalization of the second sieve for $v$-over-Ramanujan primes is based on the sequence of intervals
\begin{equation}\label{10.12}
(\lfloor2v\rfloor-\varepsilon,\lceil3v\rceil+\varepsilon),(\lfloor3v\rfloor-\varepsilon,\lceil5v\rceil+\varepsilon),
(\lfloor5v\rfloor-\varepsilon,\lceil7v\rceil+\varepsilon),\ldots
\end{equation}
with the removing intervals containing less than two primes (cf. (\ref{7.2})). For every remaining interval, we write the primes (in increasing order) except for the last one. Then all remaining primes are $v$-over-Ramanujan.
For example, if $v=3,$ we obtain the following sequence of $3$-over-Ramanujan primes (sequence A164952 in \cite{11}):
\begin{equation}\label{10.13}
2, 3, 11, 17, 23, 29, 41, 43, 59, 61, 71, 73, 79, 97, 101, 103, 107,\ldots \;.
\end{equation}
Furthermore, one can obtain a $v$-classification of primes, including $v$-left, $v$-right, $v$-central and $v$-isolated primes (see Section 8). In particular, if $l_v(n),\; r_v(n)$ denote the $n$-th $v$-left prime and the $n$-th $v$-right prime, respectively, then $l_n\sim r_n$ as $n\rightarrow\infty.$
Consider now a natural generalization of Proposition \ref{p4} with the similar proof.
\begin{proposition}\label{p8}
Let $\pi_{R_v'}(x)$ be the counting function of $v$-over-Ramanujan numbers not exceeding $x.$ Then
$$\liminf_{n\rightarrow\infty}\pi_{R_v'}(x)/\pi(x)\geq 1-\frac {1} {v}.$$
\end{proposition}
A generalization of Conjecture \ref{con1} (with a similar heuristic proof) is the following.
\begin{conjecture}\label{con2}
\begin{equation}\label{10.14}
\pi_{R_v'}(x)\sim (1-e^{-(v-1)})\pi(x).
\end{equation}
\end{conjecture}
Note that, if Conjecture \ref{con2} is true, then, using (\ref{10.2}), for the counting function $\pi_{R_v^*}(x)$ of $v$-pseudo-Ramanujan primes we have
\begin{equation}\label{10.15}
\pi_{R_v^*}(x)\sim (\frac{1}{v}-e^{-(v-1)})\pi(x),
\end{equation}
so that the proportion of $v$-pseudo-Ramanujan primes among all $v$-over-Ramanujan primes is
$(\frac{1}{v}-e^{-(v-1)})/(1-e^{-(v-1)}).$ This proportion tends to 1 as $v\rightarrow1,$
and decreases to 0 as $v\rightarrow\infty.$
Using Theorem \ref{t8}, we note that, if Conjecture \ref{con2} is true, then, for the counting function $\pi_{L_v'}(x)$ of $v$-over-Labos primes, we have
\begin{equation}\label{10.16}
\pi_{L_v'}(x)\sim \pi_{R_v'}(x)\sim (1-e^{-(v-1)})\pi(x).
\end{equation}
Furthermore, if Conjecture \ref{con2} is true, then, for the counting functions $\pi_{l_v}(x),\pi_{r_v}(x),\; \pi_{c_v}(x)$ and $\pi_{is_v}(x) $ of the $v$-left, $v$-right, $v$-central and $v$-isolated primes, respectively, of the considered classes of primes, we have
\begin{equation}\label{10.17}
\pi_{l_v}(x)\sim\pi_{r_v}(x)\sim (1-e^{-(v-1)})e^{-(v-1)}\pi(x),
\end{equation}
\begin{equation}\label{10.18}
\pi_{c_v}(x)\sim (1-e^{-(v-1)})^2\pi(x),
\end{equation}
\begin{equation}\label{10.19}
\pi_{is_v}(x)\sim e^{-2(v-1)}\pi(x),
\end{equation}
so that $\pi_{r_v}(x)+\pi_{l_v}(x)+\pi_{c_v}(x)+\pi_{is_v}(x)=\pi(x).$
\section{Other open problems}
\begin{conjecture}\label{con3}$(cf. \; Proposition \;\ref{p6}).$ There exist arbitrary long sequences of consecutive primes $p_k, p_{k+1}, \ldots ,p_m,$ such that every interval $(\frac {p_i}{2}, \frac {p_{i+1}}{2}),\; i=k,k+1, \ldots ,m-1,$ contains a prime.
\end{conjecture}
\begin{conjecture}\label{con4}$(cf. \; Proposition \;\ref{p3}).$
$\limsup_{n\rightarrow\infty} (R_n-L_n)=\infty.$
\end{conjecture}
\begin{conjecture}\label{con5}$(cf. \; Proposition \;\ref{p4}).$ There exist infinitely many peculiar primes.
\end{conjecture}
\begin{problem} For $v>1,$ to estimate the smallest pseudo-$v$-Ramanujan prime, the smallest $v$-central prime and the smallest $v$-isolated prime.
\end{problem}
\section{Acknowledgments}
The author is grateful to Daniel Berend (Ben-Gurion University, Israel) and Greg Martin (University of British Columbia, Canada) for important private communications \cite{2}, \cite{4} and very useful discussions. He is also grateful to Peter J. C. Moses (UK) for improvement the text and to the anonymous referee for very important remarks.
\begin{thebibliography}{16}
\bibitem{1} E. Bach, and J. Shallit, \emph{Algorithmic Number Theory}, MIT Press, 233 (1996). ISBN 0-262-02405-5.
\bibitem{2} D. Berend, Private communication.
\bibitem{3} S. Laishram, On a conjecture on Ramanujan primes, \emph{Int. J. Number Theory} \textbf{6} (2010), 1869--1873.
\bibitem{4} G. Martin, Private communication.
\bibitem{5} K. Prachar, \emph{Primzahlverteilung}, Springer-Verlag, 1957.
\bibitem{6} S. Ramanujan, A proof of Bertrand's postulate, \emph{J. Indian Math. Soc.} \textbf{11} (1919), 181--182.
\bibitem{7} S. Ramanujan, in
G. H. Hardy, S. Aiyar, P. Venkatesvara, and B. M. Wilson, eds.,
\emph{Collected Papers of Srinivasa Ramanujan},
Amer. Math. Soc.,
2000.
\bibitem{8} D. Redmond, \emph{Number Theory, An Introduction},
Marcel Dekker, 1996.
\bibitem{9} J. B. Rosser, The $n$-th prime is greater than $n\log n,$ \emph{Proc. Lond. Math. Soc.} \textbf{45} (1938), 21--44.
\bibitem{10} J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, \emph{Illinois J. Math.} \textbf{6} (1962), 64--97.
\bibitem{11} N. J. A. Sloane,
\emph{The On-Line Encyclopedia of Integer Sequences},
\url{http://oeis.org}.
\bibitem{12} J. Sondow, Ramanujan primes and Bertrand's postulate,
\emph{Amer. Math. Monthly}, \textbf{116} (2009), 630--635.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11N05.
\noindent \emph{Keywords: }
Bertrand postulate, Ramanujan prime, Labos prime, over-Ramanujan prime,
over-Labos prime, prime gaps.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A006992},
\seqnum{A055496},
\seqnum{A060715},
\seqnum{A080359},
\seqnum{A104272},
\seqnum{A164288},
\seqnum{A164294},
\seqnum{A164333},
\seqnum{A164368},
\seqnum{A164554},
\seqnum{A164952},
\seqnum{A166251},
\seqnum{A166252},
\seqnum{A166307},
\seqnum{A182365},
\seqnum{A182366},
\seqnum{A182391},
\seqnum{A182392},
\seqnum{A182423},
\seqnum{A182426},
\seqnum{A182451},
\seqnum{A193507},
\seqnum{A193761},
\seqnum{A193880},
\seqnum{A194184},
\seqnum{A194186},
\seqnum{A194217}, and
\seqnum{A194598}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received August 4 2011;
revised versions received September 7 2011; May 8 2012; May 16 2012.
Published in {\it Journal of Integer Sequences}, May 29 2012.
\bigskip
\hrule
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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